@shasheminassab  asked in Aggregate GPS data by zipcode  how to convert GPS coordinates to Zip Codes. @gzmorgan0  pointed out this is an impossible job to do completely correctly, for many reasons. An important reason is that Zip Codes are fixed points (that may change tomorrow) on a mail carrier's route; they may overlap and they may not cover the area and they were never intended to represent an area. The Census folks produce something close enough for some purposes, called ZCTA (zip code tabulation area) files. Here's some JSL to download the 2019 ZCTA and use it for forward and reverse geocoding. There are two tiny examples at the end. Test it before deciding it does what you need.   If( JMP Version() < "15", Throw( "JMP 15 required" ) ); zipcodeNS = New Namespace( "zipcodeNS" ); zipcodeNS:zipCodeSetup = Function( {}, {i, za, blobshp, tempfile, blobdbf, zipkeys, latlonvals}, If( !File Exists( "$desktop/cb_2019_us_zcta510_500kdbf.jmp" ) // | // !File Exists( "$desktop/cb_2019_us_zcta510_500kshp.jmp" ) // , // // from https://www.census.gov/geographies/mapping-files/time-series/geo/cartographic-boundary.html za = Open( "https://www2.census.gov/geo/tiger/GENZ2019/shp/cb_2019_us_zcta510_500k.zip", "zip" ); blobshp = za << read( "cb_2019_us_zcta510_500k.shp", Format( blob ) ); tempfile = Save Text File( "$temp/cb_2019_us_zcta510_500k.shp", blobshp ); zipcodeNS:dtshp = Open( tempfile, Add to Recent Files( 0 ), invisible ); zipcodeNS:dtshp << save( "$desktop/cb_2019_us_zcta510_500kshp.jmp" ); blobdbf = za << read( "cb_2019_us_zcta510_500k.dbf", Format( blob ) ); tempfile = Save Text File( "$temp/cb_2019_us_zcta510_500k.dbf", blobdbf ); zipcodeNS:dtdbf = Open( tempfile, Add to Recent Files( 0 ), invisible ); zipcodeNS:dtdbf << save( "$desktop/cb_2019_us_zcta510_500kdbf.jmp" ); , // zipcodeNS:dtshp = Open( "$desktop/cb_2019_us_zcta510_500kshp.jmp", Add to Recent Files( 0 ), invisible ); zipcodeNS:dtdbf = Open( "$desktop/cb_2019_us_zcta510_500kdbf.jmp", Add to Recent Files( 0 ), invisible ); ); zipcodeNS:dtdbf:shape << Set Property( "Link ID", 1 ); zipcodeNS:dtshp:shape << Set Property( "Link Reference", Reference Table( zipcodeNS:dtdbf ) ); zipcodeNS:dtshp:name( "ZCTA5CE10[Shape]" ) << hide( 0 ); zipcodeNS:dtsummary = zipcodeNS:dtshp << Summary( invisible, Group( :Shape ), Mean( :X ), Mean( :Y ), Freq( "None" ), Weight( "None" ), statistics column name format( "stat of column" ), Link to original data table( 0 ) ); zipcodeNS:dtsummary:shape << Set Property( "Link Reference", Reference Table( zipcodeNS:dtdbf ) ); // validate shape is monotonic ascending index For( i = 1, i <= N Rows( zipcodeNS:dtsummary ), i += 1, If( zipcodeNS:dtsummary:shape[i] != i, Throw( "bad shape indexing" ) ) ); // add a column that can be used with nrows to find the rows of the shape in dtshp zipcodeNS:dtsummary << New Column( "LastRow", Numeric, "Continuous", Format( "Best", 12 ), Formula( Col Cumulative Sum( :N Rows ) ) ); zipcodeNS:dtsummary << runformulas; // build the lookup table for initial guessing zipcodeNS:kdt = KDTable( zipcodeNS:dtsummary[0, {"mean of Y", "mean of X"}] ); // the forward lookup table produces a coordinate from the polygon's // mean (median is worse) coordinates that is inside the polygon 90% // of the time. zipkeys = zipcodeNS:dtsummary:name( "ZCTA5CE10[Shape]" ) << getvalues; latlonvals = As List( (zipcodeNS:dtsummary:mean of y << getvalues) || (zipcodeNS:dtsummary:mean of x << getvalues) ); zipcodeNS:aa = Associative Array( zipkeys, latlonvals ); zipcodeNS:aa << Set Default Value( . ); zipcodeNS:dtsummary << setdirty( 0 ); zipcodeNS:dtshp << setdirty( 0 ); zipcodeNS:dtdbf << setdirty( 0 ); ); zipcodeNS:zipCodeSetup(); zipcodeNS:LatLonToZip = Function( {lat, lon}, {shapeid, distance, iCandidate, testid, endrange, beginrange}, If( !Is Missing( lat ) & !Is Missing( lon ), {shapeid, distance} = zipcodeNS:kdt << KNearestRows( 10, lat || lon ); For( iCandidate = 1, iCandidate < N Cols( shapeid ), iCandidate += 1, testid = shapeid[iCandidate]; endrange = zipcodeNS:dtsummary:lastrow[testid]; beginrange = endrange - zipcodeNS:dtsummary:nrows[testid] + 1; If( In Polygon( lat, lon, zipcodeNS:dtshp[beginrange :: endrange, "x"], zipcodeNS:dtshp[beginrange :: endrange, "y"] ), Return( zipcodeNS:dtshp[beginrange, "ZCTA5CE10[Shape]"] ) ); ); // inside the bounding box of the 1st one? consider that good enough. // handles points in the water in the middle of a horse shoe shape. // there are also degenerate 2 point polys, so if withing .001 degree too.. testid = shapeid[1]; endrange = zipcodeNS:dtsummary:lastrow[testid]; beginrange = endrange - zipcodeNS:dtsummary:nrows[testid] + 1; If( distance[1] < .001 | (Min( zipcodeNS:dtshp[beginrange :: endrange, "x"] ) <= lon <= Max( zipcodeNS:dtshp[beginrange :: endrange, "x"] ) & Min( zipcodeNS:dtshp[beginrange :: endrange, "y"] ) <= lat <= Max( zipcodeNS:dtshp[beginrange :: endrange, "y"] )), Return( zipcodeNS:dtshp[beginrange, "ZCTA5CE10[Shape]"] ) ); ); Return( . );// not found ); zipcodeNS:ZipToLatLon = Function( {zip}, xy = zipcodeNS:aa[Char( zip )]; If( !Is List( xy ), {., .}, xy ); ); /*** some random tests. show(zipcodeNS:LatLonToZip( 35.79, -78.65), zipcodeNS:ZipToLatLon("27605")); show(zipcodeNS:LatLonToZip(31.832,-106.391)=="79908"); // test the round trip nbad=0; nall=0; for(i=0,i<=99999,i+=1, {lat,lon}= zipcodeNS:ZipToLatLon(i); // just enough jitter here to create an occasional round-trip error... zip=zipcodeNS:LatLonToZip(lat+randomnormal(0,.0001),lon+randomnormal(0,.0001)); if(!ismissing(lat), nall+=1; if(zip!=char(i),show(i,zip);nbad+=1;) ) ); show(nall,nbad);// median bad=3936/30575. mean bad=2438/30575 //https://gist.github.com/erichurst/7882666 test=[ 29506 34.221377 -79.650336, 29510 33.454312 -79.613121, 29511 33.975732 -79.122716, 29512 34.619033 -79.718041, 29516 34.437454 -79.634640, 29518 33.788608 -79.847975, 29519 34.018637 -79.359821, 29520 34.679486 -79.927292, 29525 34.552415 -79.540885, 29526 33.847406 -78.947093, 29527 33.785083 -79.140176, 29530 33.996055 -79.740075, 29532 34.280331 -79.869336, 29536 34.419502 -79.373048, 29540 34.377298 -79.846185, 29541 34.064949 -79.740750, 29543 34.287605 -79.265927, 29544 33.926098 -79.255583, 29545 34.163660 -78.966477, 29546 33.883845 -79.349409, 29547 34.496287 -79.334829, 29550 34.399975 -80.082806, 29554 33.704050 -79.381598, 29555 33.845583 -79.483856, 29556 33.668227 -79.763139, 29560 33.827895 -79.742142, 29563 34.353366 -79.208673, 29564 33.449386 -79.848408, 29565 34.358052 -79.499289, 29566 33.873402 -78.666957, 29567 34.560972 -79.432683, 29568 33.913616 -78.750292, 29569 34.032632 -78.911583, 29570 34.667866 -79.562707, 29571 34.135123 -79.425041, 29572 33.772568 -78.785662, 29574 34.165413 -79.260831, 29575 33.629132 -78.970412]; for(i=1,i<=nrows(test),i+=1, z=zipcodeNS:LatLonToZip(test[i,2],test[i,3]); if(num(z)==test[i,1],0,show(i,z)); ll=zipcodeNS:ZipToLatLon(char(test[i,1])); if(abs(ll[1]-test[i,2])>.02 || abs(ll[2]-test[i,3])>.02,show(ll[2]-test[i,2],ll[1]-test[i,3])); ); ***/ show(zipcodeNS:LatLonToZip( 38.897,-77.043)); show(zipcodeNS:ZipToLatLon(20006)); The LatLonToZip will convert about 1000 points per second; it uses a KDTable to find a set of 10 nearby polygons which will be checked to see if the point is actually inside. There are some degenerate cases, polys with only one edge, that don't round-trip very well.  The function is set up to cache some data tables on your desktop; the 100MB download is slow. You might want to cache them in a better place.   an old article with important details: http://web.archive.org/web/20050209030255/http://www.manifold.net/cases/zip_codes/zip_codes.html more recent answers: https://gis.stackexchange.com/questions/2682/sources-for-us-zip-code-boundaries Census starting point: https://www.census.gov/geographies/mapping-files/time-series/geo/cartographic-boundary.html   ... View more

At the end of How to use Define Class I hinted there was something interesting about JSL class instances being hard to copy. if X refers to an instance, Y=X makes both Y and X refer to the same instance. The same thing happens when you pass an instance to a user defined function: the function's value is not a copy of the instance; it refers to the same instance. So changing the instance inside the function changes it for the caller as well. If you have ever passed a matrix to a function and modified it in the function, expecting the caller's copy to be updated, you know JSL passes arguments to user functions by value. JMP passes arguments by making copies, but... Notice the matrix is copied, so changes to the matrix inside the function are made to the copy, not the original. Notice the big black reference dot is also copied, but not the object it refers to. This causes beginning JAVA programmers many headaches because they have not seen this picture. Within the function, you can do two very different things to variable a. you can call a method, like a:set(42); you can assign a value, like a=42; In case (1) you change the argyle object's value from 3 to 42. This is probably what you mean to do. in case (2), you overwrite the big black dot next to a with 42. You probably didn't mean to do that.   The following code uses objects that keep references to other objects. All of the objects have a next() method, so the objects can be composed in interesting ways. The next() method produces the next value in a sequence of values. Some objects represent a constant and always produce the same next value. Others, like the add object, have two other objects (the left side and right side of the + operator). Add:next() calls left:next() and right:next() and adds the two answers. The complete JSL is attached. Here it is with the pictures interleaved. Let's start with the simplest class. // number/matrix wrapper class Define Class( "vwrap", m_v = .; // holds the number or matrix _init_ = Method( {v}, If( Type( v ) == "Class", Throw( "vwrap is for number or matrix, not class " || Char( v << getname ) ) ); m_v = v; ); set = Method( {v}, If( Type( v ) == "Class", Throw( "vwrap is for number or matrix, not class " || Char( v << getname ) ) ); m_v = v; ); next = Method( {}, m_v; // return ); ); Vwrap is the name of the class, and it wraps a value, either a number or a matrix. It has a setter that will reject values it doesn't appreciate and a getter, named next(), to return the next value in a sequence. Because it is a bit ugly to write NewObject(Vwrap(42)), let's also make a helper function to make Vwraps. // helper function v(...) converts a scalar or matrix to a vwrap, or returns an instance unchanged v = Function( {x}, If( Type( x ) == "Class", If( !(x << Contains( "next" )), Throw( "v expects number, matrix, or one of the classes it supports, not class " || (x << getname) ) ); x; // x supports a next function, it is probably OK , // else If( !Is Number( x ) & !Is Matrix( x ), Throw( "v expects number or matrix, not " || Type( x ) ) ); New Object( vwrap( x ) ); // wrap the number or matrix ) ); v(...) will get called a lot, sometimes with a class instance, sometimes with a scalar or matrix, and it needs to return the class instance unchanged when it gets one. It might as well check that there is a next() in the class at the same time, just in case there are other classes that might get mixed up. Next, a largely boiler plate function to make a graph.  // global function gr(...) makes a graph gr = Function( {Xvar, Yvar, title, n}, {Xs, Ys, dt}, // locals Yvar = v( Yvar ); Xvar = v( Xvar ); Ys = J( n, 1, Yvar:next() ); Xs = J( n, 1, Xvar:next() ); dt = As Table( Ys || Xs, <<invisible, <<ColumnNames( {"y", "x"} ) ); Eval(// capture the invisible dt as a constant in onclose(), below Eval Expr( dt << Graph Builder( title( title ), Show Control Panel( 0 ), Show Legend( 0 ), Variables( X( :x ), Y( :y ) ), Elements( Line( X, Y, Legend( 2 ), Row order( 1 ) ), Points( X, Y, Legend( 3 ) ) ), SendToReport( Dispatch( {}, "x", ScaleBox, {Add Ref Line( 0, "Solid", "Black", "", 1 ), Label Row( Show Major Grid( 1 ) )} ), Dispatch( {}, "y", ScaleBox, {Add Ref Line( 0, "Solid", "Black", "", 1 ), Label Row( Show Major Grid( 1 ) )} ), Dispatch( {}, title, OutlineBox, {Set Title( "" )/*, Image Export Display( Normal )*/} ), Dispatch( {}, "graph title", TextEditBox, {Set Text( title )} ) ), <<onclose( Close( Expr( dt ), nosave ); 1; ) ) ) ); ); You can see the calls to v() for the X and Y parameters; those can be simple numbers that will get wrapped, or other classes with a next() method. Then the J() function makes matrices of N values, and a invisible table is made, and Graph Builder runs, with a bit of magic to close the hidden table when it is no longer needed. Let's try it. gr( 13, 42, "Value 13,42", 1 ); A single point, at x=13, y=42 Hang on, it will get better...Let's make a counter class that will return 0,1,2,3,... on successive calls to next(). // counter class Define Class( "counter", m_rate = .; // non zero increment m_pos = empty(); // internal position _init_ = Method( {rate}, m_rate = v( rate ); ); next = Method( {}, rate = m_rate:next(); // see peek() comment in "sin" class If( Is Empty( m_pos ), result = 0 * rate; // get scalar or matrix dimension on first call m_pos = rate; // for next time , // else result = m_pos; // from last tim m_pos += rate; // for next time ); result; // return ); ); The _init_ method takes a rate that determines how fast the counter increments. Rate can be a constant, typically 1, or it can be another class instance. Here's a helper function to make a counter, and a first graph. // global function to make a counter ctr = Function( {rate = 1}, New Object( counter( rate ) ) ); gr( ctr(), ctr() , "count up A (equal steps)", 5 ); Notice two separate counter instances were passed to gr(...) for the X and Y axes. Two counters, running in parallel, for five steps starting at zero. Now, for something different: let's make the counter's rate be another counter. // this variation begins to show the power of composition gr( ctr(), ctr( ctr() ), "count up B (first step is 0)", 5 ); The first step is zero because the rate counter's first result is zero. This graph's Y axis is using functional composition of two counters. Lets add a sin() class. // sin() wrapper class Define Class( "sin", m_rate = .; // 0<step<PI. m_amp = .; // amplitude m_pos = .; // internal position _init_ = Method( {rate, amp}, // m_rate = v( rate ); m_amp = v( amp ); // pos needs same dimension (or scalar) as rate, but start at 0. // This is going to use up the first value from rate prematurely // and could be rearranged like the counter class to avoid that, // but you can't really see the issue in the variable frequency // graph. I think adding a peek() might be a good choice. m_pos = -m_rate:next(); ); next = Method( {}, // amp and rate can mix'n'match scalar and matrix m_amp:next() :* Sin( m_pos += m_rate:next() ) // return ); ); Sin produces an infinite sine wave at a frequency and amplitude determined by constants or other class instances. It also keeps an internal position. You can tell from the comments that it might need more thought, but it works for this demo. Here's a single cycle waveform. gr( ctr(), New Object( Sin( 2 * Pi() / 100, 3 ) ), "sin()", 101 ); Single cycle of sine wave Similar to the counter with an increment of another counter, a sine wave can have a variable frequency. gr( ctr(), New Object( Sin( New Object( Sin( .0002, 1 ) ), 3 ) ), "variable frequency", 1000 ); Ramping up the frequency using the first bit of another sine wave And two different frequency sine waves can be plotted against each other. gr( New Object( Sin( 4 * Pi() / 99, 3 ) ), New Object( Sin( 6 * Pi() / 99, 3 ) ), "2X3 sin()", 100 ); Simple integer ratios make nice patterns Another class and helper function; the class adds one or more other classes together and the offset helper adds a constant...or another function, probably...untested. // add class Define Class( "add", m_a = {}; //m_b = .; _init_ = Method( {a, b = Empty()}, If( Is Empty( b ), For( i = 1, i <= N Items( a ), i += 1, Insert Into( m_a, v( a[i] ) ) ), m_a[1] = v( a ); m_a[2] = v( b ); ) ); next = Method( {}, result = m_a[1]:next(); For( i = 2, i <= N Items( m_a ), i += 1, result = result + m_a[i]:next() ); result; ); ); // helper function uses add to offset another value by off offset = function({off,x}, newobject(add(off,x)) ); With the offset helper one of the early graphs that had an initial zero step can be reworked to have an initial step of one. // compared to a previous example, this one increments earlier gr( ctr(), ctr( offset(1,ctr( 1 )) ), "count up C (first step is 1)", 5 ); Now all the steps are one bigger than before. // building a square wave by summing odd harmonics n = 100; basefreq = 2 * Pi() / n; freq1 = New Object( Sin( 1 * basefreq, 1 / 1 ) ); freq2 = New Object( Sin( 3 * basefreq, 1 / 3 ) ); freq3 = New Object( Sin( 5 * basefreq, 1 / 5 ) ); freq4 = New Object( Sin( 7 * basefreq, 1 / 7 ) ); sum = New Object( Add( {freq1, freq2, freq3, freq4} ) ); gr( ctr(), sum, "4 Odd Harmonics", n + 1 ); Summing odd harmonics at reduced amplitude approximates a square wave. // same square wave, a lot more odd harmonics n = 200; basefreq = 2 * Pi() / n; freqs = {}; For( i = 1, i <= 23, i += 2, Insert Into( freqs, New Object( Sin( i * basefreq, 1 / i ) ) ) ); gr( ctr(), New Object( Add( freqs ) ), "12 Odd Harmonics", n + 1 ); More harmonics help square it up   // multiply class Define Class( "multiply", m_a = {}; _init_ = Method( {a, b = Empty()}, If( Is Empty( b ), For( i = 1, i <= N Items( a ), i += 1, Insert Into( m_a, v( a[i] ) ) ), m_a[1] = v( a ); m_a[2] = v( b ); ) ); next = Method( {}, result = m_a[1]:next(); For( i = 2, i <= N Items( m_a ), i += 1, result = result :* m_a[i]:next() ); result; ); ); // multiply two waveforms n = 1000; freq1 = New Object( Sin( .1, 1 ) ); freq2 = New Object( Sin( 2 * Pi() / n, 1 ) ); prod = New Object( Multiply( freq1, freq2 ) ); gr( ctr(), prod, "multiply", n + 1 ); Multiply two frequencies; the lower frequency creates the envelope. // similar, but control amplitude of freq1 using another sin wave n = 1000; freq2 = New Object( Sin( 2 * Pi() / n, 1 ) ); // lo freq freq1 = New Object( Sin( .1, freq2 ) ); // hi freq, control amplitude with freq2 gr( ctr(), freq1, "amp is sin", n + 1 ); Alternatively, control the amplitude of the higher frequency using the lower frequency. Finally, make a vwrap holding a matrix of odd harmonic frequencies and make a new class to sum the matrix into a scalar. // try a matrix...use v(...) to make a vwrap to hold the matrix n = 200; mat = v( ((1 :: 49 :: 2) * (2 * Pi() / n)) ); // add the sin wrapper. sinmat:next() returns a row vector of 5 sin values sinmat = New Object( Sin( mat, 1 / (1 :: 49 :: 2) ) ); // matrixSum assumes the argument class is returning a matrix and returns the scalar sum Define Class( "matrixSum", m_mat = .; _init_ = Method( {mat}, m_mat = v( mat ) ); next = Method( {}, Sum( m_mat:next() ) ); ); gr( ctr(), New Object( matrixSum( sinmat ) ), "25 Odd Harmonics", n + 1 ); It will take a lot more harmonics to really reduce the over-shoot and the ripples.   Good doc: https://www.jmp.com/support/help/en/15.1/index.shtml#page/jmp/classes.shtml  ... View more

The following is mostly right; I may have missed something or misunderstood something; read the linked doc (it is too short) and fire away in the comments - thanks.    Bad analogy: a class is a cookie cutter. An instance is a cookie made from the cookie cutter.    Problem with analogy: I'm also going to call JSL's class a prototype. A prototype is an actual cookie. The prototype cookie could be eaten or modified, but you wouldn't normally want to do that. Instead, you should trace its outline into the cookie dough and cut that with scissors to make new instances of cookies. But if you do take a bite out of the prototype cookie, all future cookie instances will look bitten too (second example). Also, the _init_(...) method is not run for the prototype cookie, so it may be half-baked, but it has the right shape.   The documentation uses template rather than prototype. I prefer prototype over template; a prototype is a usable object while a template just shows where the spray paint should go. This subtle distinction does not matter for the first example.    A simple example   This is a simple single-class example for bubble. The class prototype is made by DefineClass(). NewObject() makes  61 actual instances from the prototype. Each of the 61 instances has a radius and an x,y center. The prototype also provides methods to _init_(...), drift(), and draw(). A graph is displayed using the object  draw method for each object. Then a loop runs, calling the object drift method for each object, and then telling the graph to redraw. The result is an animated graph like this:     The JSL for the graph:   // define a simple class (like a template) to represent a 2D bubble Define Class("bubble", // class member variables m_radius = .; // class member variables are specified in m_xpos = .; // the class definition with initial values. m_ypos = .; // these values are copied into instances created by NewObject(...) // class methods _init_ = Method( {radius, x, y}, // called by NewObject(bubble(r,x,y)) below to initialize an instance this:m_xpos = x; // this: is the namespace of an instance m_ypos = y; // you don't need "this:" unless the member name is m_radius = radius; // the same as a parameter name ); drift = Method( {}, // a method to move the bubble m_xpos = m_xpos * 0.995;// tentatively drift to center m_ypos = m_ypos * 0.995; ); draw = Method( {}, // a method to draw the bubble Circle( {m_xpos, m_ypos}, m_radius ) ); ); // make a bunch of instances of the bubble class, keep them in a list bubblelist = {}; For( i = 1, i <= 61, i += 1, Insert Into( bubblelist, // each instance of bubble is initialized with a random radius and position New Object( bubble( Random Uniform( .03, .8 ), Random Uniform( -.9, .9 ), Random Uniform( -.9, .9 ) ) ) ) ); picno = 0; // need an ID for each picture // make a window to display all the instances of the bubble class New Window( "bubble class instances", gb = Graph Box( // the graphic script for the graph box scales and sizes the axes and... xaxis( {Add Ref Line( 0, "Solid", "Black", "", 1 )} ), yaxis( {Add Ref Line( 0, "Solid", "Black", "", 1 )} ), framesize( 600, 600 ), X Scale( -1, 1 ), Y Scale( -1, 1 ), // the graphic script tells each bubble to draw itself For( i = 1, i <= N Items( bubblelist ), i += 1, bubblelist[i]:draw() ); text({-.9,-.9},char(picno)), <<onclose( // experimenting is simpler if the window stops the updates when the window closes keepRunning = 0; 1; ) ) ); // to make a video, set up a directory to hold the pictures dir = "$temp\BubbleClassImages_DeleteMe\"; Delete Directory( dir ); // clean up previous results, if any Create Directory( dir ); // create/re-create // the while(forever) loop does two things on each pass: drift all the instances and update the graph keepRunning = 1; // this flag is set to zero when the window is While( keepRunning, // closed, stopping this loop For( i = 1, i <= N Items( bubblelist ), i += 1, bubblelist[i]:drift() ); gb << inval << updatewindow; // tell the window it needs to redraw picno += 1; // displayed on graph and part of filename, if used // gb << save picture( dir || Char( (picno) + 1e6 ) || ".png" ); Wait( .01 ); // a wait is needed to make sure the window actually redraws ); My mental model for how it works: under the covers, DefineClass() keeps a hidden list of all the DefineClass() prototypes and they look a lot like instances. More about that in a moment; the example above accesses them through the NewObject() function. NewObject() clones a new instance of the prototype object and runs the _init_ method.    What's this:? In the example above I named the member variables with m_ prefix. That's not required, but it can be useful for both remembering what variables are member variables, and for preventing collisions with parameter names and local variable names. But I also used this:m_xpos on one of the _init_() method assignments. The this: namespace means "the current instance's member variable." So this:x = x; might be used if you want the member variable x and the parameter x to have the same name. m_ will be easier for the next person that maintains your code.   What's bubble:? There's another namespace that you mostly don't need that will show why I said the DefineClass() prototypes in the hidden list look a lot like instances--the class name is a namespace. Using bubble:m_radius will access the prototype's m_radius, even from an instance method. If you think you need a static member variable, this is a way to make one...if you are careful.   Using the prototype for static values   The next example uses a static m_ID variable. As the comments suggest, I probably should have made a different variation, but this one is still useful because it explains the "if you are careful" warning. Here's what happens (my mental model, again): When the _init_() below runs, the bubble2 class has already copied all of the prototype content into the new instance. The new instance has a copy of the prototype's m_id. _init_() then changes the m_id() in the prototype. The next instance gets the changed value. The other member functions have to choose between the m_id in the prototype (bubble2:m_id) and this:m_id (or just m_id with no namespace). Here, they use m_id which is a copy of a unique-to-the-instance integer.  In the video, the last piece of the puzzle falls into place at 5:15 at the bottom edge of the screen. The black circles are bubble2s that are displaying their m_id value. They use the m_id to make sure to ignore themselves when studying all the near neighbors.     // twist is a scaling divisor for a shearing rotation that moves // the outer edge faster. // this value needs to get bigger with more beehive cells. global:twist = 600; // choose the size to make them fit, but it also interacts with twist // because the distance from the center grows global:bubblesize = 0.057; // use a defined class to represent each cell in the beehive. Define Class("bubble2", m_radius = .; // in the video, all beehive cells have the same radius m_xpos = .; // the beehive cell's position m_ypos = .; // // this is answering the question "can a JSL Class have a static member variable?" // this could have been done more intuitively for this example by passing the // index of the creation loop to the _init_ code, but here it is...initialize the // "static" value to 1... m_id = 1; // keep reading to understand the two-faced nature of this variable... // What actually happens: when newObject(bubble2(...)) makes a new instance, the _init_(...) // is called. // First, the m_id is cloned from the prototype to the instance. The first instance gets m_id==1. // Second, the bubble2: namespace causes the prototype class to increment the // prototype's m_id. The next instance gets the new value from the prototype, m_id==2. // this is a little odd, coming from a C++ -like language; in the drift method below, // "other:m_id != m_id" both m_id values are regular non-static instance members. // if you want the static behaviour, use the bubble2:m_id approach (bubble2 is the class // name and access the singleton in the prototype class.) // Normally you *never* want to use the class name as a name space qualifier because // you set up the class correctly and initialize via _init_( parameters ). And normally // a static value is also a constant value, so using a copy is just fine. But if // multiple instances need to communicate via a singleton static, AND you always use // the namespace, this is a reasonable place to keep the static. // Having written all that, I now dislike this particular example because it is using // a confusing combination of bubble2:m_id and instance values of m_id. _init_ = Method( {radius, x, y}, m_radius = radius; this:m_xpos = x; // this: (or nothing) operates on an instance m_ypos = y; bubble2:m_id++; // bubble2: operates on the prototype. The next newobject(bubble2(...)) uses the modified prototype ); distanceBetweenCenters = Method( {other}, /*return*/ Sqrt( (m_xpos - other:m_xpos) ^ 2 + (m_ypos - other:m_ypos) ^ 2 ) ); distanceBetweenEdges = Method( {other}, // negative: overlaps /*return*/ distanceBetweenCenters( other ) - (m_radius + other:m_radius) ); drift = Method( {}, // bubbles should pull to 0,0 AND move away from near bubbles angle = ATan( m_ypos, m_xpos ); radius = Sqrt( m_ypos ^ 2 + m_xpos ^ 2 ); shear = global:twist; // also depends on number of circles shrink = .99950; // radius/shear creates a tiny amount of rotation, more on the edges. That rotational // shearing helps the circles bounce about. /400 is not enough and /100 is too much. // radius*shrink pulls all the circles to the origin. It is balanced by push, below. x = Cos( angle + radius / shear ) * radius * shrink; y = Sin( angle + radius / shear ) * radius * shrink; // check neighbors {neighbors, distances} = kdt << kNearestRows( 7, m_id ); tension = 0; ntension = 0; // calculate a tension, or stress, for contour to use as a color For( iN = 1, iN <= N Items( neighbors ), iN += 1, i = neighbors[iN]; other = bubblelist[i]; If( other:m_id != m_id, // <<<<<<<<<<<<<<<< NOT the bubble2:m_id value !!! dist = distanceBetweenEdges( other ); // if the dist is zero, it is perfect. if the other guy is far away, ignore him. If( Abs( dist ) < global:bubblesize * 0.8, tension += dist ^ 2; // this will be a RMS value at the bottom ntension += 1; ); If( dist < 0, //push angle = ATan( m_ypos - other:m_ypos, m_xpos - other:m_xpos ); x += Cos( angle ) * Abs( dist ) ^ .4 * .05; y += Sin( angle ) * Abs( dist ) ^ .4 * .05; );// , // Beep() ); ); m_xpos = x; m_ypos = y; If( ntension == 0, ntension = 1; tension += (global:bubblesize / 2) ^ 2; ); Sqrt( tension / ntension ); // RMS ); draw = Method( {}, Transparency( .8 ); Circle( {m_xpos, m_ypos}, m_radius, "FILL" ); Transparency( 1 ); Text( center justified, {m_xpos, m_ypos - m_radius * .2}, Char( m_id ) ); ); ); bubblelist = {}; n = 12; // how many "rings" of hex cells m = -26 + 21 * n + 3 * (n - 3) ^ 2; // yes, I used JMP to get this formula For( i = 1, i <= m, i += 1, Insert Into( bubblelist, New Object( bubble2( Random Uniform( global:bubblesize, global:bubblesize ), Random Uniform( -.9, .9 ), Random Uniform( -.9, .9 ) ) ) ) ); // need a table for contour plot, and it will include 100 values in a circle around the hive ring = 100; dt = New Table( "tension", Add Rows( m + ring ), New Column( "x", Numeric, "Continuous", Format( "Best", 12 ), Set Values( [] ) ), New Column( "y", Numeric, "Continuous", Format( "Best", 12 ), Set Values( [] ) ), New Column( "tension", Numeric, "Continuous", Format( "Best", 12 ), Set Values( [] ) ) ); For( i = 1, i <= ring, i += 1, angle = 2 * Pi() * i / ring; dt:x[m + i] = 1.25 * Sin( angle ); dt:y[m + i] = 1.25 * Cos( angle ); dt:tension[m + i] = 0; ); cplot = dt << Contour Plot( X( :x, :y ), Y( :tension ), Show Data Points( 0 ), Fill Areas( 1 ), Label Contours( 0 ), Transform( "Range Normalized" ), Color Theme( "Green Yellow Red" ), Specify Contours( Min( 0 ), Max( .01 ), N( 50 ) ) ); Report( cplot )[framebox( 1 )] << Frame Size( 1920, 1080 ) << maximizewindow( 1 ) << backgroundcolor( "black" ); Report( cplot )[framebox( 1 )] << xaxis( {Format( "Fixed Dec", 12, 3 ), Min( -1.4*1920/1080 ), Max( 1.4*1920/1080 ), Inc( 0.5 ), Label Row( {Show Major Grid( 0 ), Show Major Labels( 0 ), Show Major Ticks( 0 ), Show Minor Grid( 0 ), Show Minor Labels( 0 ), Show Minor Ticks( 0 )} )} ); Report( cplot )[framebox( 1 )] << yaxis( {Format( "Fixed Dec", 12, 3 ), Min( -1.4 ), Max( 1.4 ), Inc( 0.5 ), Label Row( {Show Major Grid( 0 ), Show Major Labels( 0 ), Show Major Ticks( 0 ), Show Minor Grid( 0 ), Show Minor Labels( 0 ), Show Minor Ticks( 0 )} )} ); // here is the script that makes the bubbles draw Report( cplot )[framebox( 1 )] << addgraphicsscript( Fill Color( "black" ); text Color( "white" ); pen Color( rgbcolor(64,64,64) ); For( i = 1, i <= N Items( bubblelist ), i += 1, bubblelist[i]:draw() ); hline(0); vline(0); ); cplot << onclose( keepRunning = 0; Close( dt, nosave ); 1; ); Wait( 0 ); //dir = "$temp\ClassWithStatic" || Char( global:twist ) || "\"; //Delete Directory( dir ); //Create Directory( dir ); //Write( Convert File Path( dir, "windows" ) ); picno = 0; keepRunning = 1; While( keepRunning, xy = J( N Items( bubblelist ), 2, . ); For( i = 1, i <= N Items( bubblelist ), i += 1, xy[i, 1] = bubblelist[i]:m_xpos; xy[i, 2] = bubblelist[i]:m_ypos; ); // the kdtable makes it faster to find near neighbors kdt = KDTable( xy ); For( i = 1, i <= N Items( bubblelist ), i += 1, dt:tension[i] = bubblelist[i]:drift(); dt:x[i] = bubblelist[i]:m_xpos; dt:y[i] = bubblelist[i]:m_ypos; ); Report( cplot )[framebox( 1 )] << inval << updatewindow; picno += 1; // Report( cplot )[framebox( 1 )] << save picture( dir || Char( (picno) + 1e6 ) || ".png" ); Wait( .01 ); );   Both examples above could just as easily (maybe easier) have been built with rows of a data table to represent objects and columns to represent member variables and JSL user functions for the methods. The use of classes, especially without any subclassing, might make your JSL harder for someone else to follow if they are not comfortable with classes in JSL. Speaking of Functions, the doc for DefineClass mentions them too. As far as I can figure, they bring almost nothing to the table and should probably be avoided. I think every instance of a class is probably getting a separate copy of every method and every member variable. If you define a Function() and store it in a member variable, you are just making something else to copy that (maybe) could have been in a more global namespace with only a single copy. But if there will only be a single (or very few) instances and you really need a function for the class that is unaware of anything except its parameters, well maybe a function in a class is what you need. Moving on to the final example, let's do some subclassing and look at the last namespace, super:, which I had to hunt for (see comments in JSL).   Intuitive example: a base class might be a vehicle, subclasses (or derived classes) are trucks and cars. A truck might replace the base class navigation method with one that knows to avoid residential streets. A car might take advantage of HOV lanes. A list of vehicles will each do the right thing when told to navigate from A to B.   Define Class has an interesting argument: baseclass(...). In the example below, shape will be the base class and square and circle will be derived from shape. The example makes a list of squares and circles and displays them. two squares, two circles, two outlined, two filled The following description is based on trial and error; I believe it is correct enough to be a helpful model to predict how something will work.   What's super:? When a derived class is created, JMP runs to the end of the baseclass chain and works back to the top, creating base classes first, and in their own namespaces. The namespaces are linked together internally so that a derived class can reference a member variable by looking in its own namespace, then following the namespace chain down to the base class until finding the variable. The super: namespace just means skip forward once in the namespace chain without looking in the current namespace. That's why the JSL below is saying "don't use super: to call a base method if you don't need to"; the base method won't have access to the derived namespace so it can't call a derived function. Multiple inheritance: if you are using "interfaces" and avoid duplicate names, it might work. Good luck. I'm not sure where a sibling baseclass fits into the chain of namespaces in the model above.   // demonstrates JMP class inheritance and overriding a method // and calling the super class. A circle and a square class // are derived from a shape class. Explains when to NOT use super:. // // I found super: in https://github.com/sassoftware/jsl-hamcrest/blob/master/Source/Core/Matchers/EqualTo.jsl // it isn't documented anywhere else, but works great for calling base class _init_ // Also, https://github.com/sassoftware/jsl-hamcrest/blob/master/Source/Core/Utils.jsl // shows a "self" implementation idea. This is a bad idea because it creates a circular reference // that must be cleared before the object is lost/deleted...or it will leak. // avoid making members and methods in base and derived classes with the same name. // _init_, _to string_, _show_ are exceptions, but they must not try to use a derived // class member or method. (you would not normally want to, so no problem.) // // avoid needless upcast (super:); it cripples the base class because there is no downcast or // virtual mechanism. Without the upcast, a baseclass function runs with access to the // derived class members and methods. (see square's draw method calling docolor calling record.) // // it appears that the derived chain is searched *after* the current class; in a two class // (base,derived) example that inverts the search order for members and methods when a // derived class uses "super:" to call a base class method. If the base class calls a "virtual" // function that exists in both base and derived, it won't get the derived one if "super:" was // used but will if no scope is used. You only need "super:" typically when you want to call // the base class method from an overridden method of the same name. If the base class method // isn't calling back to derived via a virtual, everything is good. // deep details // // the output shows lines like this: // record square=square(shape=(s= x=30 y=30 r=5 s=4 c=yellow f=1)) // record circle=circle(shape=(s= x=30 y=60 r=10 s=99 c=green f=0)) // // the square and the circle both get the right answer via different // paths. Because of way draw() is overridden by square and not by circle, // the initial call, objects[window:i] << draw(), is calling either a // derived member or a base member. In both cases, "this" is for the // derived member. in square's draw(), I made an error, now commented // out: /*super:*/docolor(), that caused docolor to run with "this" // for the base class, not the derived class. docolor then calls record(). // all three classes have record() members. The intent is to override the // base class version ("never seen"). The error caused the base class // version to run for square because "this" of docolor() was for the base class. Define Class( "shape", // base class for both shapes m_Xcenter = .; m_Ycenter = .; m_radius = .; m_sides = .; m_color = .; m_filled = .; _init_ = Method( {x, y, radius, sides, color, fill}, m_Xcenter = x; m_Ycenter = y; m_radius = radius; m_sides = sides; m_color = color; m_filled = fill; ); _to string_ = Method( {}, "shape=(s=" || " x=" || Char( m_Xcenter ) || " y=" || Char( m_Ycenter ) || " r=" || Char( m_radius ) || " s=" || Char( m_sides ) || " c=" || Char( m_color ) || " f=" || Char( m_filled ) || ")" ); _show_ = _to string_; // the following should NOT ever get called because square and circle override it // try changing square to call super:docolor and then you'll see this record=method({},show("never seen, overridden by square and circle")); // both shapes understand filled vs outlined // see note in square:draw method about calling docolor without super: docolor = Method( {}, this:record(); // this: doesn't change anything, If( this:m_filled, // it is mostly for disambiguating Fill Color( /*this:*/m_color );// parameter of same name , // Pen Color( this:m_color ) ) ); // the base class has a draw method that circle uses; // square provides an override because this draws a // diamond-oriented square draw = Method( {}, docolor(); p = J( m_sides, 3, 2 );//2==draw p[1, 3] = 1;//1==move p[m_sides, 3] = -2;//-2==draw and close For( i = 1, i <= m_sides, i += 1, p[i, 1] = m_Xcenter + m_radius * Cos( 2 * Pi() * i / m_sides ); p[i, 2] = m_Ycenter + m_radius * Sin( 2 * Pi() * i / m_sides ); ); Path( p, m_filled ); ); ); Define Class( "circle", baseclass( shape ), _init_ = Method( {x, y, radius, color, fill}, super:_init_( x, y, radius, 99/*sides*/, color, fill ) ); _to string_ = Method( {}, "circle(" || super:_tostring_() || ")" ); _show_ = _to string_; record=method({},write("\!nrecord circle=",_show_())); // circle has no override for draw() ); Define Class( "square", baseclass( shape ), _init_ = Method( {x, y, radius, color, fill}, super:_init_( x, y, radius, 4/*sides*/, color, fill ) ); _to string_ = Method( {}, "square(" || super:_tostring_() || ")" ); _show_ = _to string_; record=method({},write("\!nrecord square=",_show_())); // square overrides the base class draw draw = Method( {}, /*super:*/docolor();// don't up-cast, docolor can't down-cast and gets base class record() Rect( m_Xcenter - m_radius, m_Ycenter - m_radius, m_Xcenter + m_radius, m_Ycenter + m_radius, m_filled ); ); ); objects = {}; Insert Into( objects, New Object( square( 30, 30, 5, "yellow", 1 ) ) ); Insert Into( objects, New Object( Circle( 30, 60, 10, "green", 0 ) ) ); Insert Into( objects, New Object( square( 60, 30, 15, "blue", 0 ) ) ); Insert Into( objects, New Object( Circle( 60, 60, 20, "red", 1 ) ) ); New Window( "objects", gb = Graph Box( framesize( 500, 500 ), For( window:i = 1, window:i <= N Items( objects ), window:i += 1, //Show( objects[i] ); objects[window:i] << draw(); // uses the base class or the derived class draw ) ) ); //stop(); // animate the color of object 4, the big red circle For( i = 0, i <= 1, i += (1 / 256), //show(i); objects[4]:m_color = HLS Color( i, .5, 1 ); gb << inval << updatewindow; Wait( 0 ); );   What about this? C++ has a value named "this" (other languages use "self" and "me") that is sometimes needed for an object to refer to itself. Not the same as this: namespace prefix. I don't think the JSL class offers a "this" within the class, and you might be tempted to make one like this: x = newObject(myClass(17)); x::self = x; or something close to that. Then a myClass method could use "self" to do do things that require a class instance, such as cloning. Scripting index showing messages that can be sent to a class instance Keeping a member variable in a class instance that contains the (pointer to) class instance can create a memory leak. This brings us to a new topic...topics actually.   When is an instance deleted? An instance is deleted when there are no references to it. In the section above, a "self" reference inside an instance will prevent the instance from being deleted, even if you've lost all your handles to the instance. There is no way to clean it up, other than closing JMP.   When is an instance copied? Here's the best part: an instance is only copied if you use the <<clone method! As far as I can tell, there is no other way to copy an instance. This means instances are passed by reference. Yay! Way less clumsy to make a simple class to wrap a large matrix and pass an instance of the class to a user function, instead of having JMP make a copy of the matrix argument into the parameter value, and possibly returning another copy.   ... View more

I've been working on a video project for a while, and finally ran into a wall with VLC. VLC is the best video player. It can also be used as a video generation tool, but has some quirks related to FPS of the output files. I decided to re-investigate FFmpeg as a replacement. I'd used FFmpeg a long time ago, without really reading all the docs and remembered it being complicated. Well, it is. I spent today reading the docs. It is a command line tool with tons of doc (see link) . I put this together as a proof-of-concept. There's a lot more it can do. The JSL below consists of a user defined function that generates images for a video sequence and a RunProgram call that runs FFmpeg to make a video. FFmpeg is told to get images via stdin, and RunProgram provides them by calling the user defined function. I have only run it on Windows, but I believe it will only take some minor changes to make it run on Mac as well. To run it, you'll need to download FFmpeg. Their website points to another site you can get it from. I'm not sure why they don't have an official windows version. Once installed, this script will run in about 30 seconds to make an 8-second video, then open the video in your default player. It will look better in VLC, or you can upload it to youtube and see it like this:   Read the legal stuff carefully if you are going to do commercial work with it.   // generate images for ffmpeg stdin on the fly to make a video // using https://ffmpeg.org/download.html#build-windows // // You could make a somewhat simpler version, maybe, by writing // the files to disk and telling ffmpeg to re-read them. I wanted // to use the stdin pipe, and didn't try a file. I hope this pipe is // at least as fast as writing and reading the files. // // you can't make a 2nd, parallel stream of audio data using RunProgram, // so you'll need a disk file for audio. Or http as shown... outputmp4 = ConvertFilePath("$desktop/VideoMadeByJMPandFFMPEG.mp4", "windows"); nrows = 1080 / 2;// keep the image buffer even in both dimensions ncols = 1920 / 2;// and maybe a multiple of 16 as well Open Log( 1 ); // where the action is (may need this window open for the text box to work) imagex = J( nrows, ncols, RGB Color( 1, 1, 1 ) ); // buffer of JSL color values, updated by getNextImage... // credit the CC music...write this once, into the buffer musicCredit = Text Box( "\!"Pump Sting\!" Kevin MacLeod (incompetech.com) Licensed under Creative Commons: By Attribution 4.0 License http://creativecommons.org/licenses/by/4.0/ "||char(asdate(today())), <<setwrap( ncols ), <<backgroundcolor( RGB Color( 1, 1, 1 ) ), <<setfontsize( 16 ), <<fontcolor( "black" ) ); // grab the pixels from the text box... creditPixels = (musicCredit << getpicture) << getpixels; // and stuff them into the buffer imagex[(N Rows( imagex ) - N Rows( creditPixels ) + 1) :: (N Rows( imagex )), (1 + 20) :: (20 + N Cols( creditPixels ))] = creditPixels; nn = 0; // what ever state information is needed. here, just a counter. getNextImage = Function( {}, If( nn <= 255, Try( imagex[// make another little square block on each call Interpolate( nn, 0, 1, 256, nrows ) :: Interpolate( nn + 1, 0, 1, 256, nrows ), // Interpolate( nn, 0, 1, 256, ncols ) :: Interpolate( nn + 1, 0, 1, 256, ncols )// ] = HLS Color( nn / 256, .5, 1 );// color of block changes nn += 1; // update counter/state info Matrix To Blob( -imagex, "int", 4, "big" ); // return the argb data (negative JSL color) , Show( exception_msg ); // oops. what went wrong? Empty(); // shut down after error ) , Empty() // normal shutdown after all images delivered ) ); start = Tick Seconds(); // rp will hold the RunProgram object; below there is a loop to wait for it to complete rp = Run Program(// downloaded from https://ffmpeg.zeranoe.com/builds/win64/static/ffmpeg-20200626-7447045-win64-static.zip executable( "C:\Users\v1\Desktop\ffmpeg-20200626-7447045-win64-static\bin\ffmpeg.exe" ), options( {// idea from https://stackoverflow.com/questions/34167691/pipe-opencv-images-to-ffmpeg-using-python "-y", // overwrite output "-f", "rawvideo", // tell ffmpeg the stdin data coming in is "-vcodec", "rawvideo", // "raw" pixels, uncompressed, no meta data "-s", Eval Insert( "^ncols^X^nrows^" ), // raw video needs this meta data to know the size of the image "-pix_fmt", "argb", // the JSL color values are easily converted to argb with MatrixToBlob "-r", "20", // input FPS "-i", "-", // this is the pipe input. everything before this is describing the input data //"-an", // disable audio, or supply a sound file as another input "-i", "https://incompetech.com/music/royalty-free/mp3-royaltyfree/Pump%20Sting.mp3", // "-vcodec", "mpeg4", // encoding for the output "-b:v", "5000k", // bits per second for the output outputmp4 // the name of the output } ), WriteFunction( // the WriteFunction is called when FFMPEG wants JMP to write some data to FFMPEG's stdin Function( {this}, data = getNextImage(); // getNextImage provides the binary data for an image, or empty() when done If( Is Empty( data ), // this << writeEOF;// tell FFMPEG there is no more data , //else this << Write( data );// send the data );// ) ), ReadFunction( // the ReadFunction is called when FFMPEG wants JMP to read some of FFMPEG's stdout status messages Function( {this}, While( rp << canread, Write( this << read ); // JMP will copy them to the JMP log window Wait( .01 ); // get as much as we can so it doesn't pile up or leave some behind ) ) ) ); // If you had something else to do, you could do it before this wait loop. FFMPEG closes the // read pipe after sending its last message, the message is processed by the ReadFunction above. While( !(rp << isreadeof), Wait( .01 ) );// wait(.01) responds in 1/100 second without burning much CPU. stop = Tick Seconds(); Write( "\!ndone, used ", Char( stop - start, 7, 1 ), " seconds" ); // without this, it is hard to be sure... open(outputmp4);   ... View more

Similar to Recursive Tree Generator , but a bit more organic. This implementation uses Bezier curves to outline the trunk, limbs, and twigs. When the twigs are divided too finely, a clump of leaves is added. The JSL path(...) function is used to draw the Bezier trunk and the leaves, though the leaves are just an octagon. When a path is filled, and it crosses over itself, it does an even-odd count to determine if the fill color is applied. With the twigs this isn't really desirable, but with the leaves it makes a nice effect. Lots of leaf octagons with odd/even overlap Internally, the algorithm creates a random shaped tree, following the central spine of each trunk/limb/twig and recording the Bezier points for the left side on the way to the leaves and the Bezier points for the right side on the way back to the trunk. The point that is on the center of a segment is a drawn point; the two control points on each side are where the segment meets other segments. Spine, bark, control points for Bezier Changing the minimum limb size controls how deep the recursion goes; some of these turned off the leaves for clarity. Just a trunk. The end cap is rounded with a Bezier too.   This branch point chose a three-way split.   Add the octagon leaves. They don't overlap because the size was adjusted.   One more level.   The overlapping leaves show the odd/even effect of crossing paths.   Tiny leaves on a big tree.   Tiny, still more branching   Big leaves, lots of branches.   // Recursive routine builds a random tree using Bezier curves in path(...) command TreePath = Function( {x0, y0, size0, angle},// parameters for recursion {x0L, x0R, y0L, y0R, x1, y1, size1, x1L, x1R, y1L, y1R, nbranches, leftangle, rightangle, ibranch, nbranches, angles, length, biggest, biggestv, middle, middlev, cx0, cy0, cx1, cy1, cxc, cyc, xcleaf, ycleaf, radleaf, xleaf, yleaf, diameter, leafangle}, // local variables // x0,y0 is the center of this trunk segment, starting point. angle is the direction, and size0 the radius. leftangle = angle + Pi() / 2; // find two points at right angles to the direction of growth rightangle = angle - Pi() / 2; // by adding +/- 90 degrees ... this is all radians, so PI/2 x0L = x0 + size0 * Cos( leftangle ); // size0 is the radius y0L = y0 + size0 * Sin( leftangle ); x0R = x0 + size0 * Cos( rightangle ); y0R = y0 + size0 * Sin( rightangle ); length = size0 ^ 1 * Abs( Random Normal( 30, 2 ) );// scaling constants...play with these size1 = .95 * size0;// .95 is a limb shrinkage constant. Not sure this is enough to notice the taper. x1 = x0 + length * Cos( angle ); // x1,y1 is the other endpoint of this trunk segment y1 = y0 + length * Sin( angle ); x1L = x1 + size1 * Cos( leftangle ); // and similar right angle calculation y1L = y1 + size1 * Sin( leftangle ); // to find the bark at the other end x1R = x1 + size1 * Cos( rightangle ); y1R = y1 + size1 * Sin( rightangle ); // the center of the trunk segment is ON the drawn curve. the endpoints become // control points that pull the curve toward the joints, partway, leaving smooth bends. Insert Into( global:pathTrunk, Eval List( {Eval List( {x0L, y0L, 0 /*control*/} )} ) ); Insert Into( global:pathTrunk, Eval List( {Eval List( {(x0L + x1L) / 2, (y0L + y1L) / 2, 3/*bezier*/} )} ) ); Insert Into( global:pathTrunk, Eval List( {Eval List( {x1L, y1L, 0 /*control*/} )} ) ); If( size0 > .5, // the recursion stops when the branch becomes twigs; leaves are added then <grin> nbranches = Random Integer( 2, 3 ); // experiment with number of branches here angles = J( nbranches, 1, Abs( Random Normal( 4, 1 ) ) ); // I decided the trees generally looked better if the bigger branch was centered... biggest = Loc Max( angles ); biggestv = angles[biggest]; middle = Ceiling( N Rows( angles ) / 2 ); middlev = angles[middle]; angles[middle] = biggestv; angles[biggest] = middlev; angles = angles / Sum( angles ) * Pi();//normalize, radians // repurpose leftangle and rightangle, using previous leftangle as a starting point: // there is a semicircle at the end of this branch, pi() radians (180 deg), going through // x1L,y1L and x1R,y1R which form a diameter. The first branch leaves from x1L,y1L and // the rightangle on the semicircle (computed inside the loop.) The angle is half way, // and the x0y0 is on the chord, halfway. When this routine is recursively called, // it will recompute the two points on the chord from the angle and size, which is half the chord length. For( ibranch = 1, ibranch <= nbranches, ibranch += 1, // working on a branch...figure out its base rightangle = leftangle - angles[ibranch]; // x1,y1 is the semicircle center. get the chord endpoints at the left and right angles and radius size. cx0 = x1 + size1 * Cos( leftangle ); cy0 = y1 + size1 * Sin( leftangle ); cx1 = x1 + size1 * Cos( rightangle ); cy1 = y1 + size1 * Sin( rightangle ); // and the midpoint of the chord cxc = (cx0 + cx1) / 2; cyc = (cy0 + cy1) / 2; // and the diameter of the branch (length of chord) diameter = Sqrt( (cx0 - cx1) ^ 2 + (cy0 - cy1) ^ 2 ); // here it is! do the child branch (which will do its child...) Recurse( cxc, cyc, diameter / 2, (leftangle + rightangle) / 2 ); // set up for next branch... leftangle = rightangle; ); , //else cap this branch and add the leaves // the cap is a rounded line across the end instead of more branching Insert Into( global:pathTrunk, Eval List( {Eval List( {x1L, y1L, 0 /*control*/} )} ) ); Insert Into( global:pathTrunk, Eval List( {Eval List( {(x1L + x1R) / 2, (y1L + y1R) / 2, 3/*bezier*/} )} ) ); Insert Into( global:pathTrunk, Eval List( {Eval List( {x1R, y1R, 0 /*control*/} )} ) ); // add leaves to the twigs...circle diameter twig length, centered on twig... // the single path with all these circles makes an interesting pattern when the circles overlap // and cancel each other out in an even/odd way. The trunk does that too, and there seems to be no way to prevent it. xcleaf = (x1L + x1R) / 2; // center a sphere of leaves over the cap segment ycleaf = (y1L + y1R) / 2; radleaf = 100 * size1; // constant to play with - radius of leaf ball around cap's size For( leafangle = 0, leafangle < 2 * Pi(), leafangle += Pi() / 4, xleaf = xcleaf + radleaf * Cos( leafangle ); yleaf = ycleaf + radleaf * Sin( leafangle ); // move to the new leaf ball, then draw the ball Insert Into( global:pathLeaves, Eval List( {Eval List( {xleaf, yleaf, If( leafangle == 0, 1/*move*/, 2/*draw*/ )} )} ) ); ); ); // ditto, x0R,y0R to x1R,y1R is the trailing (right side) edge. But reversed. Insert Into( global:pathTrunk, Eval List( {Eval List( {x1R, y1R, 0 /*control*/} )} ) ); Insert Into( global:pathTrunk, Eval List( {Eval List( {(x0R + x1R) / 2, (y0R + y1R) / 2, 3/*bezier*/} )} ) ); Insert Into( global:pathTrunk, Eval List( {Eval List( {x0R, y0R, 0 /*control*/} )} ) ); ); // these two globals are filled in by the recursive routine global:pathTrunk = {}; global:pathLeaves = {}; size = 11; // of the trunk, radius really treepath( 0, 0, size, Pi() / 2 ); // do the recursion // leading move, control...this adds the left-side ground Insert Into( global:pathTrunk, {{-size, 0, 0}}, 1 );//push control Insert Into( global:pathTrunk, {{-size - 5, 0, 1}}, 1 );//push move // trailing control, draw...this adds the right-side ground Insert Into( global:pathTrunk, {{size, 0, 0}} );//push control Insert Into( global:pathTrunk, {{size + 5, 0, 3}} );//push move // convert the lists into matrices. It was easier to add to the lists during recursion // but path() needs the matrix matTrunk = Matrix( global:pathTrunk ); matLeaves = Matrix( global:pathLeaves ); New Window( "Example", Graph Box( X Scale( -600, 600 ), Y Scale( -100, 1100 ), framesize( 950, 950 ), <<backgroundcolor( HLS Color( .66, .2, 1 ) ), // dark blue Fill Color( HLS Color( .16, .3, .5 ) ); // brown Path( matTrunk, 1 ); Transparency( .75 ); // play with this... Fill Color( HLS Color( .33, .6, .8 ) ); // green Path( matLeaves, 1 ); ) ); It is a little hard to figure out what the path() function expects for a Bezier curve, but it makes sense when it is done. The first item in the list should be a move (1), and the last item should be a Bezier point (3).  Two control points (0) appear between each pair of Bezier points and between the initial move and Bezier. The drawn line goes through the move (1) and Bezier (3) points. The control points (0) determine the shape of the curve; the line does not go through them but does go toward them. The code above makes the pair of control points both the same point (blue, way above). There must still be two. See this answer for code to experiment with behavior. It is a graph.   Bézier is the correct spelling. Pronunciation is harder for me. https://en.wikipedia.org/wiki/B%C3%A9zier_curve  ... View more